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Stream: theory: category theory

Topic: Yoneda structures for fibrations

Matteo Capucci (he/him) (Mar 29 2023 at 13:45):

Echoing this message from the nForum.
Did anyone work out the Yoneda structure on fibred categories? In even more generality, is ${\bf Fib}_{\cal K}(A)$ (for $A: \cal K$ and $\cal K$ a 2-category equipped with a Yoneda structure) equipped with an induced Yoneda structure?

Matteo Capucci (he/him) (Mar 29 2023 at 13:47):

Related (I think @Christian Williams might know the answer to this): does the 2-category of fibred categories extend to a proarrow equipment? I guess proarrows should be 'fibred profunctors' of some sort.

Matteo Capucci (he/him) (Mar 29 2023 at 13:50):

(I guess a fibred profunctor between $p:E \to B$ and $q:D \to B$ is a profunctor $f:E \to D$ which is 'over the hom-profunctor of $B$', i.e. when you turn them into spans the apex of the one obtained from $f$ is fibred over the apex obtained from $B(-,=)$)

Nathanael Arkor (Mar 29 2023 at 13:52):

I think the paper you want to look at is Street's "Conspectus of variable categories".

Matteo Capucci (he/him) (Mar 29 2023 at 13:58):

Thank you :D

Matteo Capucci (he/him) (Mar 29 2023 at 14:03):

FYI it seems fibrational cosmoi are also of interest to me

Christian Williams (Mar 29 2023 at 16:03):

Categories, functors, profunctors, and two-sided fibrations form a triple category. The squares are transformations, fibered functors, and fibered profunctors; the cube is a fibered transformation.

(There's an explanation on my thread.)

So yeah, restricting to one half of this triple category, then fibrations form an equipment which is doubly fibered over Cat.